Gravity Illustrated

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Gravity IllustratedSpacetime Edition

RICKARD JONSSON

Department of Theoretical PhysicsAstronomy and Astrophysics group

Chalmers University of Technology and Goteborg UniversityGoteborg, Sweden 2004

Gravity IllustratedSpacetime Edition

Rickard JonssonISBN 91-7291-519-6

c! Rickard Jonsson, 2004.

Doktorsavhandlingar vid Chalmers Tekniska Hogskola,Ny serie nr 2201ISSN 0346-718x

Department of Theoretical PhysicsAstronomy and Astrophysics groupChalmers University of Technology and Goteborg UniversitySE-412 96 GoteborgSwedenTelephone +46(0)31-772 10 00

Cover: An illustration of the curved spacetime for a line through Earth. The smalltoy car with the pen is used to draw straight lines on the spacetime – correspondingto the motion of three apples thrown out from the center of the earth. The yellow linecorresponds to the motion of the yellow apple visible outside Earth. The figure wascreated using Matlab and Povray.

Back cover: An illustration of the spacetime for a line outside Earth, and the space-time trajectory of an apple that has been thrown upwards. The figure was createdusing Matlab.

Gravity IllustratedRickard M. Jonsson

Department of Theoretical PhysicsAstronomy and Astrophysics group

Chalmers University of Technology and Goteborg University

AbstractThis thesis deals with essentially four different topics within general relativity: ped-agogical techniques for illustrating curved spacetime, inertial forces, gyroscope pre-cession and optical geometry. Concerning the pedagogical techniques, I investigatetwo distinctly different methods, the dual and the absolutemethod.In the dual scheme, I start from the geodesic equation in a 1+1 static, diagonal,

Lorentzian spacetime, such as the Schwarzschild radial line element. I then find an-other metric, with Euclidean signature, which produces the same geodesics x(t). Thisgeodesically equivalent dual metric can be embedded in ordinary Euclidean space.Freely falling particles correspond to straight lines on the embedded surface.In the absolute scheme, I start from an arbitrary Lorentzian spacetime with a given

field of timelike four-velocities uµ. I then perform a coordinate transformation to thelocal Minkowski system comoving with the given four-velocity at every point. In thelocal system the sign of the spatial part of the metric is flipped to create a new metricof Euclidean signature. For the particular case of two dimensions we may embedthe absolute geometry as a curved surface. The method is well suited for visualizinggravitational time dilation, cosmological expansion and black holes.Concerning inertial forces, gyroscope precession and optical geometry, the gen-

eral framework is based on the introduction of a congruence of reference worldlinesin an arbitrary spacetime. This allows us to describe the local motion and accelera-tion of particles in terms of the speed relative to the congruence, the time derivativeof the speed and the spatial curvature (project down along the reference congruence)of the corresponding worldline.I present two papers concerning inertial forces in this framework, one formal and

one intuitive. I also present two papers concerning gyroscope precession, again oneformal and one intuitive. In particular I illustrate how one can explain gyroscopeprecession in an arbitrary stationary spacetime as a double Thomas precession effect.Introducing a novel type of spatial curvature measure for the worldline of a test

particle, we present a natural way of generalizing the theory of optical geometry toinclude arbitrary spacetimes. The generalized optical geometry allows us to do opti-cal geometry across the horizon of a black hole.

Keywords: curved spacetime, embeddings, pedagogical techniques, inertial forces,gyroscope precession, optical geometry

APPENDED PAPERS

Paper IEmbedding spacetime via a geodesically equivalent metric of Euclidean signatureR. Jonsson – Gen. Rel. Grav. (2001) 33 no 7, pp. 1207-1235.

Paper IIVisualizing curved spacetimeR. Jonsson – Accepted for publication in Am.J.Phys.

Paper IIIInertial forces and the foundations of optical geometryR. Jonsson – To be submitted to Class. Quant. Grav.

Paper IVAn intuitive approach to inertial forces and the centrifugal force paradoxR. Jonsson – To be submitted to Am.J.Phys.

Paper VA covariant formalism of spin precession with respect to a reference congruenceR. Jonsson – To be submitted to Class. Quant. Grav.

Paper VIAn intuitive derivation of spin precession in stationary spacetimesR. Jonsson – To be submitted to Am.J.Phys.

Paper VIIGeneralizing optical geometryR. Jonsson, H. Westman – To be submitted to Class. Quant. Grav.

Paper VIIIOptical geometry across the horizonR. Jonsson – To be submitted to Class. Quant. Grav.

AcknowledgementsFirst of all I would like to express my gratitude towards my supervisor, Marek Abra-mowicz for allowing me to work on what I find interesting – and to develop as anindependent researcher. I would also like to thank Ulf Torkelsson and SebastianoSonego for all sorts of comments and advice.Imust of course also acknowledge the sole Bohmian-Mechanics-Lover of the house

– who is also my friend, colleague and fellow rebel in physics – Hans Westman. Wehave shared more discussions and laughs (many at the expense of various physicists)than I can remember. ’Wha Wha Wha ...’:)While Achim Tassemark didn’t include me in his acknowledgements, I am a for-

giving man and would not in any way hold that against him:) He has been my com-panion in angling, longbow archery, smithery, horseback riding and much else overthe years and likely will be in the future as well.It can be debated whether they deserve it, but I would like to acknowledge also

’the most unlikely creatures of all’ – the Lord of the String PhD-students below us (inevery respect). May the spirit of Gert Jonnys guide you for ever:)I would also like to mention the very best longbow shooting nephews of mine

that the world has ever seen – David and Alexander:)More important than anyone I must acknowledge Agneta – whose name is still

magic, and who taught me that physics is just crap – compared to what matters inlife.Lastly I would like to acknowledge my parents, among many other things for

making sure that I had something to eat while finishing this thesis. They are, as TinaTurner puts it ’Simply the Best’ (although I very much doubt that Tina was referringto her parents:).

September, 2004

Rickard Jonsson

Grow strongerFrom the 13’th warrior

Hope is for freeFabrizio

CONTENTS

1 Introduction 11.1 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 An introduction to Einstein’s gravity 52.1 The spacetime diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Flat spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Mass curves spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Curved spacetime for a

staff outside Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 The spacetime for a line

through the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6 Forces and gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 About mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.8 More about mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 Was Newton wrong and

was Einstein right? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Making illustrations 153.1 About matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Some shortcomings of matlab . . . . . . . . . . . . . . . . . . . . 173.2 About povray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Matlab to povray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Making models 214.1 The funnel shaped spacetime . . . . . . . . . . . . . . . . . . . . . . . . 214.2 The bulgy cylinder spacetime . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Spacetime visualization 275.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1.1 A technical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.1 A technical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

vii

6 Inertial forces 316.1 A technical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7 Gyroscope precession 337.1 A technical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8 Optical geometry 358.1 Generalized optical geometry in technical terms . . . . . . . . . . . . . 36

9 Conclusion and outlook 37

10 A spherical interior dual metric 4110.1 Conditions for spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4110.2 The dual interior metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 4210.3 Approximative internal sphere . . . . . . . . . . . . . . . . . . . . . . . 4310.4 Spheres in the Newtonian limit . . . . . . . . . . . . . . . . . . . . . . . 43

11 The absolute visualization 4511.1 The perch skin intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . 4511.2 Vacuum field equations for flat absolute metric . . . . . . . . . . . . . . 4711.3 On geodesics and flat metrics . . . . . . . . . . . . . . . . . . . . . . . . 4811.4 Closed dimensions and timelike loops . . . . . . . . . . . . . . . . . . . 4911.5 Warp drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5011.6 Finding generators to make a single trajectory an absolute geodesic . . 51

11.6.1 Considering generators . . . . . . . . . . . . . . . . . . . . . . . 5211.6.2 Considering photons . . . . . . . . . . . . . . . . . . . . . . . . . 52

12 Kinematical invariants 5512.1 The definitions of the kinematical invariants . . . . . . . . . . . . . . . 5512.2 The average rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

12.2.1 A matrix formulation of rotation . . . . . . . . . . . . . . . . . . 5712.3 About deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.4 Back to four-dimensional formalism . . . . . . . . . . . . . . . . . . . . 59

12.4.1 The four dimensional analogue to the rotation vector . . . . . . 60

13 Lie transport and Lie-differentiation 6113.1 Contravariant Lie differentiation . . . . . . . . . . . . . . . . . . . . . . 6113.2 Covariant Lie differentiation . . . . . . . . . . . . . . . . . . . . . . . . . 6413.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

viii

1Introduction

Ever since it was presented in 1916, Einstein’s General theory of Relativity concern-ing space, time and gravitation, has been extremely successful in explaining all sortsof gravitational phenomena. Two examples from the beginning of the century arethe gravitational deflection of light from distant stars passing close to our sun andthe precession of the perihelion of Mercury. More recent experiments involve grav-itational redshifts, relativistic slowing down of atomic clocks and indirect measure-ments of gravitational waves.Since the theory was presented it has also puzzled the minds of physicists and

people in general. Indeed, for most people the theory is still clouded in mystery. Inparticular, people find the legendary black holes fascinating yet incomprehensible.To understand Einstein’s theory onemust understand its heart – the curved space-

time. Unfortunately it is easier said than done to explain this concept without usingmathematics. There are however ways of working around the difficulties and two ofmy scientific papers are directly related to this.In fact the whole process of explaining relativity has become something of a pas-

sion for me. I have spent countless hours doing computer-generated illustrationsand more time than I care to remember down in the workshop making real modelsof curved spacetime. Using these models and illustrations I have given over thirtylectures on General Relativity in university courses as well as at high schools, sciencefestivals and conferences.I believe there is plenty of room for improvement in general of how we teach

physics. The author of a book, whether popular or scientific, may have a crystalclear understanding of the field he writes about. As a reader translates the writtenwords into his own mental language, a lot may be lost however. We can minimizethis loss by using examples, working on the exact formulations and rewriting the textaccording to how well it was understood. Still, much of the intuition that we haveconcerning physics is encoded as mental images and movies. I believe the whole

1

2 1. Introduction

learning process can become much more efficient if we actually create these imagesand movies. Of course it takes a lot of effort to produce interesting and pedagogicalillustrations. With the aid of better computers and software – our opportunities to doso have however improved. For fields like General Relativity, for which there is evena large interest from the general public, it might be worth the effort.I hope to one day write a book on General Relativity directed towards a general

audience. The idea is to use large color images spanning over double pages, to useno mathematics (or maybe just a little, well hidden in an appendix :) and to useanalogies. I believe that one can give a very good understanding of General Relativityin this manner for anyone who is interested in the mysteries of Einstein’s theories.The second chapter of this thesis gives an idea of what I want to achieve, althoughthe layout and form of this thesis is limiting to say the least.While all my work is related to General Relativity, it is not all related to popu-

larizing the field. I have also worked on inertial forces and gyroscope precession. Anexample of an inertial force is the apparent force pushing us outwards as we take asteep curve at high speed with a car. A gyroscope is essentially a rapidly spinningbody, that tends to keep its direction of spin. In general relativity we can howevermake the gyroscope turn or precess just by moving it around. For each of these twofields I have written both a formal paper and a paper more directed towards intuitiveunderstanding.I have also worked on the field of optical geometry. In brief, the idea behind this

theory is to consider a rescaled (stretched) version of the standard spacetime. Relativeto the rescaled spacetime some effects, like gyroscope precession, can be explainedin a more straight-forward manner. In one paper I show, together with my colleagueHans Westman, how one can generalize the standard theory of optical geometry soit applies to a wider class of spacetimes than the standard theory of optical geometrydoes. In yet another paper I show how one may apply this generalization to consideroptical geometry also for the inside of a black hole.

1.1 This thesis

This thesis serves many purposes apart from a being a means of collecting the ap-pended papers. One purpose is to present mywork, and the general field of work, fora more general audience. Another purpose is to present results that are not includedin the papers. Lastly, the thesis provides an opportunity to review some topics thatare underlying the appended papers. The thesis is organized in the following way:

• Chapter 2 gives an introduction to General Relativity, aimed at a general audi-ence. The main object of this chapter is to explain how spacetime geometry canexplain why things are falling towards the Earth when we drop them.

1.1. This thesis 3

• Chapter 3 gives a brief introduction to the practical side of making illustrationsusing computer software like Matlab and PovRay.

• Chapter 4 gives an account of how I made the real models that I use whenlecturing on General Relativity.

• Chapter 5 is a brief introduction to the topic of pedagogical techniques withingeneral relativity, and my work therein.

• Chapter 6 introduces the reader to inertial forces and mywork within this field.

• Chapter 7 is an introduction to gyroscope precession and my work related tothis.

• Chapter 8 gives an introduction to optical geometry and my two papers con-cerning this.

• Chapter 9 provides some brief conclusions and an outlook.

Then follows a part of the thesis that contains technical comments on the papers. Twochapters are introducing new insights and results concerning Papers I & II, and twochapters are reviews.

• Chapter 10 is a comment to Paper I, regarding the shape of the spacetime insidea star or a planet. I show that in the Newtonian limit it corresponds to an exactsphere, as depicted on the front page of this thesis.

• Chapter 11 is a comment to Paper II where I present some additional resultsand insights that are not included in Paper II.

• Chapter 12 reviews the kinematical invariants defined for a reference congru-ence of worldlines. This chapter is related to Papers III-VIII.

• Chapter 13 reviews Lie transport and Lie differentiation. This chapter is mainlyrelated to Paper III, although it also has some relevance for Papers IV-VIII.

Then follow the appended papers.

4 1. Introduction

2An introduction to Einstein’s gravity

Gravity is something that we are all more orless accustomed to. If we throw an apple up-wards, or maybe of bowling ball, we know thatit will soon fall down again.One might wonder why everything that we

throw upwards insists on falling down again.The two most renown theories for this wereput forward byNewton (late 17’th century) andby Einstein (early 20’th century).

Newton’s theory explains the fact that theapple returns by a gravitational force pullingthe apple back towards Earth.In Einstein’s theory there is no gravitational

force. Instead it explains the motion of the ap-ple by a law saying that the motion correspondsto a straight line in a curved spacetime. To explainwhat this really means is the purpose of thischapter. Let us however warm up by explain-ing what a spacetime diagram is.

2.1 The spacetime diagramImagine that you are living on a straight staffwith blue position markings on it (Fig. 2.1). On

the staff a number of events happen: an alarmclock rings, a fire cracker explodes and a walk-ing man puts down his feet.

Ring!

Ring!Ring!

Poff!

Tap!

Figure 2.1: Life on a staff.

We can mark the events happening on thestaff in a spacetime diagram (Fig. 2.2). Thelater the event happens – the higher up in thediagram it is marked. If the event is far to theright along the staff, it is marked to the right inthe diagram.

Ring!

Ring!

Ring!

Poff!

Tap!

Tip!

Tip!

0m 1m 2m 3m 4m 5m 6m 7m0s

1s

2s

3s

4s

5s

6s

7s

Time

Position

Figure 2.2: The spacetime diagram. Here we canmark everything that happens on the staff. Can wededuce from the diagram in what direction themanis moving?

5

6 2. An introduction to Einstein’s gravity

In the diagram we can also illustrate motionof objects by so called worldlines (Fig. 2.3). Thehigher the velocity – the more tilted the world-line (if straight up is considered as not tilted).

0m 1m 2m 3m 4m 5m 6m 7m0s

1s

2s

3s

4s

5s

6s

7s

Time

Position

Figure 2.3: Worldlines describing motion. Theleftmost line corresponds to an object moving to-wards the left along the staff, the middle to an ob-ject at rest and the rightmost to an object that is firstat rest but is then accelerated to high velocity to-wards the right.

Soon we will discuss the concept of curvedspacetime illustrated as a more or less curvedsurface. The flat spacetime diagram can thenbe regarded as a map of the curved spacetime,similar to how a map can illustrate the curvedsurface of Earth.

2.2 Flat spacetimeWenowplace the staff somewhere out in emptyspace, where we do not notice any effects ofgravity (Fig. 2.4).

Poff!

0.0 m 0.2 m 0.4 m 0.6 m 0.8 m 1.0 m 1.2 m 1.4 m 1.6 m

Poff!

Poff!

Figure 2.4: A staff in space. Along the staff vari-ous events happen and objects are moving.

In Einstein’s theory we have here a so calledflat spacetime which we can illustrate by a flatplane (Fig. 2.5). On the planewe describe eventsandmotion along the staff in space, just like wedid earlier in the spacetime diagram.

Time

Position

Figure 2.5: A flat spacetime for a staff in space.The upright blue lines correspond to fix positionalong the staff. What type of motion does the twoworldlines correspond to?

If we throw an apple along the staff in space,and let the apple move freely, it will continuewith constant velocity along the staff. In Ein-stein’s theory the motion of the apple is explai-ned by a law saying that the motion corres-ponds to a straight line in the spacetime.

The motion of a thrownfree object corres-

ponds to aline in the spacetime

straight

By free we here essentially mean that noth-ing is touching the object. In section 2.4 we willgive it a more precise meaning.To create a straight line on our flat space-

time surface, we could use a ruler. We willhowever instead use a little toy car equippedwith a downwards directed pen (see the coverof this thesis). The toy car, from now on de-noted the drawing car, we can then by hand rollforward. The point of using the drawing car isthat it also works on curved surfaces, as will beused later.If we know the initial velocity of a thrown

apple, we can use the drawing car to predicthow the apple is going to move. We put thedrawing car at a point on the spacetime sur-face corresponding to the starting position of

2.3. Mass curves spacetime 7

the apple and the time that we threw the apple.We give the drawing car a direction relative tothe surface, corresponding to the starting ve-locity of the apple, and roll the car forward.The drawn line corresponds to the motion ofthe apple after the throw.As an example we may predict the motion

of two apples, thrown with different velocitiesfrom the zero-meter position at zero time (Fig.2.6). Can we see were the apples are predictedto be after 0.4 seconds?

Initial directions corres-ponding to the veloci-ties 1! m/s och 3 m/s.

The straight lines we get, co-rresponding to the motion ofthe apples along the staff.

Figure 2.6: Straight lines in a flat spacetime. Thelines can be created with a three-wheeled toy car -equipped with a pen.

Fig. 2.7 illustrates how the scenario withthe apples would look in reality.

0.0 s

0.1 s

0.2 s

0.0 m 0.2 m 0.4 m 0.6 m

1,5 m/s1,5 m/s

3 m/s3 m/s

Figure 2.7: Apples thrown along a staff in space.The images show the position of the apples at con-secutive times. Compare with Fig. 2.6.

We see how one, at least in outer space, canpredict themotion of thrown apples using a ge-ometric model.

2.3 Mass curves spacetimeWe can create a curved spacetime from the flatspacetime we just showed by stretching andbending it. In Einstein’s theory spacetime iscurved bymass. The greater themass the greaterthe curvature (Fig. 2.8).

Figure 2.8: Mass curves spacetime. The Earth il-lustrates in what direction the curving mass lies.Strictly speaking the mass should lie within thespacetime – not outside as depicted here.

Sometimes one illustrates how mass curvessomething by putting a metal ball on a rubbersheet and look at how the sheet bends. Unfor-tunately this analogy tells nothing about howthe spacetime is curved which is what we wantto illustrate here. Before we talk more abouthowmass curves the spacetimewewill explainwhat it means that it is curved.

2.4 Curved spacetime for astaff outside Earth

Wenowplace the staff upright outside of Earth.Here themass of the Earth has curved the space-time as illustrated by the funnel-shaped sur-face of Fig. 2.9.Time is directed clockwise around the fun-

nel (as seen from above) and one circumferencecorresponds to one second. An object movingup along the staff at a fixed speed correspondsto a worldline spiraling around the funnel. Thefaster the object moves, the steeper the spiral.An object at rest corresponds to a horizontalcircle.


Time

Position

Poff!

Poff!

0.0m

0.2m

0.4m

0.6m

0.8m

1.0m

1.2m

Poff!

Figure 2.9: A curved spacetime for a line outsideEarth. Strictly speaking the surface should not closein on itself in the time direction. Rather one shouldcome to a new layer after one circumference – ason a paper roll, illustrated on the back cover of thisthesis. The Earth is shown in miniature.

In Einstein’s theory, here as well as in outerspace, the motion of thrown free objects corre-sponds to straight lines in the spacetime. Byfree wemean that gravity alone determines themotion of the objects (no air resistance for in-stance). We can thus predict the motion of thethrown objects by the same method that weused earlier (with the drawing car), althoughthe spacetime surface is now curved.As an example, we study two apples thrown

upwards from the zero-meter marking on thestaff with different velocities (Fig. 2.6).

3 m/s

1,5 m/s

0.0 m

0.2 m

0.4 m

Figure 2.10: Apple-throwing. As soon as the ap-ples have left the throwing hand, the motion corre-sponds to straight lines in the spacetime.

Initial directions corres-ponding to the veloci-ties 1! m/s och 3 m/s

The straight lines that we get,corresponding to the motionof the upwards thrown ap-ples along the staff.

Figure 2.11: Straight lines in a curved spacetime.Note that the light-colored (green) initial directioncorresponds to a precisely twice as high velocity asthe dark-colored (red) initial direction. Time is di-rected clockwise as seen from above.

We thus place the drawing car on the baseof the funnel, corresponding to the starting po-sition on the staff. We give it initial directionsrelative to the funnel, corresponding to the ini-tial velocities of the apples, and roll it forwardalong the surface of the funnel (Fig. 2.11).We see that both worldlines correspond to

apples that initially are moving up along thestaff, reach a maximum height, and then fallback again. The light-colored (green) line reachesa higher level on the funnel and takes a longertime before returning to the base of the funnel.This is reasonable since the corresponding ap-ple is thrown upwards with a higher velocity.You may note precisely how high the applesare expected to come, and what time it takesthem to return to their initial position. In Fig.2.12 we illustrate the motion of the apples inreality.We thus have a geometrical model for pre-

dicting the motion of apples. Unlike in New-ton’s theory there is no gravitational force inthis model. It is the shape of spacetime that de-termines that the worldline returns to the baseof the funnel, just like an upwards thrown ap-ple returns to the surface of the Earth.If the spacetime outside of the Earth would

2.5. The spacetime for a line through the Earth 9

0.0m

0.2m

0.4m

0.0 s 0.1 s 0.2 s 0.3 s 0.4 s

Figure 2.12: Upwards thrown apples along a staffoutside of Earth. The pictures show the position ofthe apples at consecutive moments in time – com-pare with the predictions from Fig. 2.11.

have a shape as illustrated to the left in Fig.2.13 – apples thrown upwards would not re-turn according to the theory. Note also that ifwe let the drawing car turn – it will not pre-dict the motion of upwards thrown free objects(illustrated to the right in Fig. 2.13.

Figure 2.13: Left: An alternative shape of space-time. A drawing car rolled straight forward wouldfollow the dashed lines that never return to thebase of the spacetime. Right: An alternative world-line. A drawing car that turns created the solid linewhich is not returning to the base of the spacetime.

Thus the shape of the spacetime and the ruleabout straight lines determine that the upwardsthrown apples should return to Earth – in Ein-stein’s theory.

In the introduction we mentioned that wewould attempt to explain the meaning of a str-aight line in a curved spacetime, and how this canexplainwhy an upwards thrown apple falls do-wn again. Hopefully this feels rather naturalnow.

2.5 The spacetime for a linethrough the Earth

To give another example of a curved spacetimewe imagine a hole straight through the Earth,with a staff that follows the hole (Fig. 2.14).

Poff!Poff!

Poff!

Figure 2.14: A staff through Earth. Along the staffevents happen and objects move.

The curved spacetime for the staff is illus-trated in Fig. 2.15. Notice that the outer partsof this spacetime correspond to the funnel thatwe earlier displayed.

Time

Position

Figure 2.15: Curved spacetime for a staff throughEarth. In this illustration it is about 8 minutes be-tween the time lines (running along the surface)and about 1500 km between the position lines (go-ing around the surface).

We nowmove to the center of the Earthwherewe simultaneously throw three apples, with dif-ferent velocities, along the staff as illustrated inFig. 2.16.This time it is perhaps not as evident what

is going to happen. But we know that as soon


Figure 2.16: Apples in the center of the Earth. Thedark-colored apple is just released and the othersare thrown with the velocities 6 and 18 km/s.

as the apples have left the throwing hand theirmotion will correspond to straight lines in thespacetime.

If we have an actual model of this type ofspacetime we can repeat the procedure withthe drawing car. We put the drawing car onthe spacetime model at a point correspondingto the center of the Earth and a certain time.We give the drawing car three different initialdirections corresponding to the three differentvelocities. The result is shown in Fig. 2.17.

Initial directionscorresponding tothe initial velocities The straight lines that we

get, corresponding tothe motion of the up-wards thrown apples

Figure 2.17: Straight lines in a curved spacetime.Note that the straight lines correspond to motionalong the staff through the Earth.

The dark (red) line corresponds to an appleat rest at the center of the Earth, the semi-light(green) line corresponds to an apple that is os-cillating back and forth along the staff, aroundthe center of the Earth. The light (yellow) linecorresponds to an apple that passes the surfaceof the Earth and continues onward into outerspace without ever returning.

These scenarios correspond precisely to whatwe would expect from reality (although an ac-tual experiment would be difficult to carry outin practice). Once more we see how straightlines in a curved spacetime can explain the mo-tion of thrown apples.As we move further away from the Earth

the funnels at the ends of the spacetime willassume a cylindric shape. But cylinders are infact flat in the sense that we can roll out a cylin-dric surface to a flat surface. When we are farfrom the curving effects of the Earth we thushave a flat spacetime, just like in the preceed-ing section.In this chapter we have illustrated different

parts of the spacetime in a certain order. All ofthese parts are however connected and there isthus only one spacetime (per universe:).

2.6 Forces and gravityIn Einstein’s theory there is no gravitational force,but there are forces also in this theory. An ex-ample is the force by which we affect a car aswe push it forward. A force acting on an ob-ject causes the worldline of the object to curve.The greater the force the greater the curvatureof the worldline.

The motion of an objectaffected by a force

corresponds to aline in the spacetime

curved

As an example we study an apple that atfirst is just hovering near a staff in outer space.The apple then receives a push towards the rightand goes off along the staff at a constant speed.How this scenario looks relative to the flat space-time is illustrated in Fig. 2.18.The law about forces and the curvature of

worldlines applies also when the spacetime iscurved. As an example we study an apple thatwe hold by its shaft here at Earth. In Einstein’stheory there is only an upwards directed forcefrom the hand acting on the apple.How the apple can remain at rest even though

2.7. About mass 11

I

II

III

I

Figure 2.18: Apple-pushing in outer space. Theworldline deviates from the straight dashed linesas the force of the push acts on the apple.

Newton’s theory Einstein’s theory

Figure 2.19: An apple held at rest at Earth. InNewton’s theory there is a gravitational force act-ing on the apple. In Einstein’s theory there is nogravitational force.

it is affected by a net force upwards may ap-pear pardoxical – but the solution is given bythe curved spacetime (Fig. 2.20).If we were to direct the drawing car along

a horizontal position line and roll it forward,it would have to turn upwards to remain at thesame height (the front wheel should be turnedto the right). The apple is thus at all times af-fected by an upwards force, just like the draw-ing car is all the time turning in the upwardsdirection – but it does not get upwards becausethe spacetime is curved!To make the apple go upwards we must act

on it by a greater force than that required tokeep it at rest. The drawing car must thus turnmore upwards than what is required to keep itat a fix height. The result would be a worldlinespiraling up along the funnel.If we on the other hand were to drop the

An upwards direc-ted force is actingon the apple...

... and makes the worldlinecurve upwards. Comparewith the straight dashed lines.

Figure 2.20: The resolution of the paradox. Anapple at rest at a height of 0.4 meters. The forcemakes theworldline curve upwards all the time butthe shape of the spacetime insures that it will notget any higher anyway.

apple, such that no forces are acting on it, theworldline would straighten up (Fig. 2.21).

Figure 2.21: An apple being released. As soonas the apple is released the worldline follows astraight line.

Note that there is no gravitational force pu-lling the apple down after we have released it.The apple falls because we cease to curve itsworldline and let it follow a straight line.

2.7 About mass

The effect of forces in Einstein’s theory is tocurve worldlines. How much a worldline is tocurve for a certain force depends on the massof the object. The greater the mass the lesserthe curvature of the worldline (Fig. 2.22). Onemight say that a large mass makes the draw-ing car hard to steer – so it tends to roll straightahead unless a great force acts on the object.


1 kg8 kg

Figure 2.22: Apple pushing in space – again. Theworldline of the heavy apple is curved less by anequal force.

That it is difficult to affect the motion of anobject with a largemass is common to both Ein-stein’s andNewton’s theory. The difference liesin that large mass does not mean large gravi-tational force in Einstein’s theory – there is noforce of gravity there.

2.8 More about massApart from the effect that mass has on the cur-vature of worldlines, it has also the propertythat it curves spacetime itself. The point is thatmass contains energy, and all sorts of energycurve the spacetime. Even a radio signal emit-ted by a cellular phone contains energy andwill curve the spacetime a little.So it is not quite as simple as that an up-

wards thrown apple corresponds to a straightline in a spacetimewhose shape is independenton the apples motion, but rather its a straightline in a spacetime that the apple itself partiallycurves (or has curved). In the case of the apple,the mass is however so small that it it does notcurve the spacetime significantly. If we on theother hand would let for instance the moon falltowards the Earth (and the Earth towards themoon), we would have to take into consider-ation that the Earth and the moon consist ofparticles that all curve the spacetime a little.The spacetime would also for this case corre-spond to an unmoving surface (it would makeno sense to have a moving spacetime), but the

shape would be more complicated than thosewe have shown in this article. It would not berotation symmetric. On the irregularly shapedsurface we would however still be able to usethe drawing car to predict the motion of an ap-ple thrown from the Earth towards the moon.

2.9 Was Newton wrong andwas Einstein right?

We have seen, although in a simplified form(that nevertheless gives precisely the right pre-dictions), how Einstein’s theory explains themo-tion of apples by a curved spacetime. Newtonon the other hand explained it with a gravita-tional force.Onemight thenwonderwhowas right? Both

theories describe with good accuracy all sortsof every day gravitation, like how a droppedapple falls towards the ground.There are however situations where the pre-

dictions fromNewton’s theory differ from thoseof Einstein’s. A famous example regards theorbit of Mercury around the sun (Fig. 2.23).

Figure 2.23: The orbit of Mercury. The dashedline is themotion as prescribed byNewton’s theory,the solid by Einstein’s. The effect that the ellipseis rotating (in the plane of the paper) is howeverexaggerated half a million times.

Measurements of the position of Mercuryfrom the 18’th century and onwards show thatthe almost elliptical orbit of Mercury is in factrotating in accordance to Einstein’s theory.So Newton’s theory does not describe real-

ity correctly. It is however simple to use for

2.9. Was Newton wrong and was Einstein right? 13

calculations compared to Einstein’s theory, andworks very well in most cases. But does thismean that Einstein’s theory about the curvedspacetime is right?When it comes to physics at a fundamental

level, one can never know if a theory is right.Tomorrow the applesmay fall upwards andwesee no way to explain this in Einstein’s theory.Then we must search for a new theory that ex-plains both why the apples fell down yesterdayand why they are falling up today.How nature reallyworks – we do not know.

What we can say for sure is that the curvedspacetime exists in Einstein’s theory –which seemsto be a good description of nature.

Figure 2.24: Curved spacetime in practice.Demonstration using a spacetime funnel and draw-ing car.


3Making illustrations

The images of this thesis have mainly been created using Matlab, under the Linuxoperating system. Post processing has been made in Corel Photo Paint. Images builtfrom several different images or vector graphics have beenmade in Corel Draw, gimpor xfig.There are considerably more advanced programs for making three-dimensional

graphics than matlab. The ideal would likely be ’Maya’ – the program that was usedto make the animated movie ’Shrek’ among others. Learning some basic tricks andbuilding your own specialized graphic functions in matlab and utilizing a freewarecalled povray, one can still come a long way.

3.1 About matlabTo create the images of this article I have written over 170 matlab functions and pro-grams of a total of more than 350 kb. To list them all would fill this entire thesis. Justto give a feel for how the various codes are working I give a little example of a matlabcode utilizing some of my functions on the next page.The non-standard matlab functions that I call here are rjxsurf, putonsurf and

rjsurfset. The function rjxsurf plots the surface much like the standard matlabfunction surf does. The main difference lies in that all the lines of the vertexes ofthe surface need not be plotted but for instance one may plot every other line byincluding ’linesep1’,2 in the arguments. To use this feature one must also sendthe viewing angles az and el. What rjxsurf does is that it puts the lines out asordinary lines (using plot3), but then it lifts them a bit towards the eye – so they arevisible on top of the surface. This function allows for the creation of smooth surfacesthat are not completely riddled with too dense-lying lines. Also setting lightingphong smooths the surface nicely.

15

16 3. Making illustrations

clfN=50;[X,Y,Z]=sphere(N);C=ones(size(X));az=160;el=20;

%%%% Plot the sphere with coordinate lines %%%%%%%%%%

rjxsurf(X,Y,Z,C,’setlines’,’on’,’az’,az,’el’,el,’linesep1’,2,...’linesep2’,2,’col’,[0 0 1],’linecol’,[1 1 0]);

%%%% Set out a ’surface line’ on the sphere %%%%%%%%%%

phibase=linspace(0,2*pi,N+1); % A coordinate basevectorthetabase=linspace(0,pi,N+1); % A coordinate basevectorphi=linspace(2.8,5.2,200); % Trajectory phitheta=pi/2+sin(phi*10).*(phi-min(phi))*0.2; % Trajectory theta

rjcell{1}={’surfline’,theta,phi,’col’,[1 0 1],’epsilon’,0.1,...’arrow’,’curved’};handles=putonsurf(X,Y,Z,thetabase,phibase,{az,el,[]},rjcell)rjsurfset(handles,{0.7,0.3,0.4})

%%%% Fixing some lighting etc %%%%%%%%%%

view(az,el);axis off;axis imageshading flatlight;lighting phong

Figure 3.1: The output from the matlab program above

3.1. About matlab 17

Printing to a file at high resolution and resampling (compressing) the image whileapplying anti-aliasing can give high quality images. The function putonsurf is es-pecially well suited for plotting worldlines on parameterized surfaces. The functionrjsurfset is just a quick way to set some reflection properties of surface elements.

3.1.1 Some shortcomings of matlabMatlab does most of what one wants concerning graphics, but it lacks some featuresthat would be useful:

• Doing real raytracing, i.e allowing objects to cast shadows and reflections onother objects. In Matlab there is only a cruder shading of surfaces dependingon the angle of the surface relative to the light source - but independent on thereflections and shadows of other surfaces.

• Doing anti-aliasing while rendering. ’Rendering’ is the process whereby a two-dimensional image is formed from a three-dimensional scenario. Anti-aliasingis a certain averaging technique of the color of nearby pixels, taking away the’staircase’ appearance (pixelization) of the edges of lines and surfaces and mak-ing them smooth.

• Doing high resolution animation. This is perhaps the most surprising deficitof matlab. It has a simple structure for making animations (movies). There ishowever no real way to control the resolution of the movie – matlab appears totake a so called screen-shot of every image. In general the screen resolution (andsize) are not what one might desire for a reasonably nice (anti-aliased) moviethat covers the screen of a standard computer or TV.

• Doing subtractions of volumes. For instance one may want to subtract a cylin-der from a sphere to make a hole through the sphere. This type of operationone must do ’by hand’ in matlab – and it can be quite time consuming andcomplicated.

Concerning the second point there is a solution. Matlab can print to a file at a veryhigh resolution, for instance issuing the command:

print -dtiff -r900 filename

The dimensions of this image is 8"6 inches. At 900 dots per inch this is a very largeimage (typically several tens ofMb). Then one opens the resulting file filename.tifin some program for manipulating two-dimensional images (Corel Photo paint inmy case). Then one resamples the image to 600 dpi and to the size that one desires,making sure that the anti-alias check-box is checked. The required resolution of the


original file to get a good final result depends on the dimensions of the final file. Asa rule of thumb one should have at least twice as many pixels (per dimension) in theoriginal image as one does in the final image, to get a nice anti-aliasing.

3.2 About povrayThere are graphics programs that have the abovementioned features that matlablacks. In particular there is a program called PovRay, a freeware downloadable fromhttp://www.povray.org. I advice the interested reader to go there and have a lookat the ’Hall of Fame’ images. Some are really spectacular in appearance, and stillcomparably easy to make.PovRay works similar to Matlab, but it is specifically designed to make images

rather than to do calculations. Here is an example of a piece of povray code in a filecalled example.pov:#include "colors.inc"background {color Cyan}camera {location <0, 2, -3>look at <0, 1, 2>}

sphere {<0, 1, 3>, 1texture {pigment{color Red}finish {reflection {0.5} phong 0.7 phong size 10 ambient 0.0 diffuse0.8}}}

plane { <0, 1, 0>, -1pigment {checker color Green, color Blue}}

light source { <2, 4, -0.001> color White parallel point at<0,0,0>}

To process (render) this file one may write (on the Unix or Linux system):

povray +P +A +W1200 +H900 example.pov

The image is then rendered and the resulting file is output as example.png. If onedesires another format, like the postscript file displayed below in Fig. 3.2, one opensexample.png in some program (like gimp or Corel Photo Paint) and converts it tothe desired format.There are however also downsides to povray compared to Matlab. For instance

one cannot interactively rotate and zoom the three-dimensional scenario using the

3.3. Matlab to povray 19

Figure 3.2: An example of the output from a short povray program.

mouse. Also the matlab standard representation of surfaces as matrixes (which is notstandard in povray) is quite practical if one is dealing with parameterized surfaces,as I have been to a large degree.

3.3 Matlab to povrayI have written a converter (a matlab script) that creates a povray file from a mat-lab generated three dimensional scene . The converter is called rjmat2pov.m andis very simple to use. After running your matlab script to create a three dimen-sional image, type rjmat2pov(’filename’) and you will get a povray file calledfilename.pov that you may render as any other povray file. What rjmat2povdoes is that it finds all the visible surface elements in the matlab scene, and dividesthem into the natural triangles that they are made of and makes a povray mesh2object out of them. The colors and reflection properties are also translated althoughthe translation of the reflection properties is rather approximate (different types ofreflection properties are used). Any light sources are also translated from the matlabscene to the povray scene. There is plenty of room for improvement of this code.Unless there are very many surface objects and large colormaps it should work justfine however. I used this converter for the cover illustration.

3.4 CommentsI plan to put thematlab files required for the examples above onmy homepage. Thereare quite a few help-files required, but so long as you put the whole pack in yourpersonal matlab directory (assuming Linux or Unix) so that matlab can find them –this need not concern you. My current homepage is at http://fy.chalmers.se/˜rico andthe files can be found by clicking the matlab symbol.


4Making models

A picture may be worth more than a thousand words, but a model is worth morethan a thousand images (or at least more than a couple of images:).If for no other reason than that it was a whole lot of work to make them, and

that I would like to be the first to write a thesis on relativity containing phrases like’turning-lathe’, ’welding’ and ’molding form’, I will now briefly explain how I madethe models that I use while lecturing about gravity.

4.1 The funnel shaped spacetimeFrom aluminum sheet metal (5 mm thick) I cut out 10 equal stripes whose shape Ihad calculated and printed (using a computer). I then waltzed these stripes to givethem the right curvature in the direction along the funnel. Next I made several press-forms of different radii and pressed the the stripes in the press-forms to give them theright curvature in the direction around the funnel. Then I TIG-welded together thepieces, see Fig. 4.1. Next I built a rack consisting of an axle and several wooden discsthat fitted snuggly to the inside of the funnel. Using the rack I turned the spacetimeround and smooth as best as possible with the lathe cutting steel. (I am skippingthe parts where I cut through the surface and had to weld it together again:) Then,while spinning it fast in the lathe, I used an angle grinder with a disc of segmentedgrinding papers to further smoothen the surface. After that followed finer grindingand polishing in the lathe. In the end it shone like a mirror.I then left it to a car lacquerer, Mr Istvan Papp, who lacquered it with a grey

metallic and clear varnish (two component).Next I put the funnel into the lathe, spun it slowly, and used a water based felt

tip pen in stead of a cutting steel to make the blue circles around the spacetime. I cutthe tip of the pen (and spent quite a few pens) to make the lines of varying width

21

22 4. Making models

Figure 4.1: A spacetime in the making. To the left the spacetime has just been spot-welded together. To the right the spacetime after turn lathing and angle grinding

(narrower towards the top of the funnel).For the black time lines along the funnel, I used masking tape and paintbrush.

For the second and meter markers I used sticker letters, the part that remains afteryou have taken out the actual letter, and used a paint brush to make the letters andnumbers.Between applying lines and text one had to put a layer of clear varnish to protect

what was already made. I am still indebted to Istvan for helping me out with this. Atlast I lacquered the inside myself with a blue metallic and clear varnished it.

4.2 The bulgy cylinder spacetimeFor the second model I used a different technique. First I glued together boards ofwood to form a big lump, with an metal axle running through the middle of it. ThenI turned and ground this big chunk of wood to a shape corresponding to half thespacetime, see Fig. 4.2.Tomake it as smooth as possible I clear varnished, waxed and polished thewooden

plug. Then I applied gelcoat (a plastic), layers of glass fibre and liquid plastic to makea glass fibremolding form. In this form Imolded the two ends of the spacetime, againusing glass fibre and liquid plastic. If you want to try this – do not neglect the useful-

4.2. The bulgy cylinder spacetime 23

Figure 4.2: From a lump of wood to a glass fibre spacetime.

ness of a really good gas mask. After cutting and grinding the ends of the spacetimeflat, I molded a means of attachment with bolts between the two ends. I also didsome molding at the ends of spacetime allowing me to tightly fit a couple of turnedaluminum discs there. Through central holes in these discs I put a steel axle enablingme to put the hole construction in the lathe and to grind away the small differencesin radius at the intersection.Then came the Time of Lacquering - that I did myself this time. Below is just the

briefest outline of a few things that went wrong in the primary tries.

• Metallic paint too old.

• Air holes in glass fibre becoming visible during lacquering.

• Partially clogged up jets on the paint sprayer.

• Wrong distance from sprayer to target.

• Dust particles on the surface getting trapped under the paint.

• Dust particles in the air sticking to the drying surface.

• Paint running from a jet getting thrown out on the surface.

• Water drops falling on un-dried paint.

24 4. Making models

• Trying to remove paint from a failed lacquering attempt - thethinner eats through the lower layers of paint creating acanyon in the surface.

• The fire department arriving, clearing the entire house,because the lacquering fog set of the alarm.

• The color appearing patchy upon rotating the surface - althoughit looked fine when not rotating it.

Then I built a scaffolding for the entire spacetime allowing me to rotate it. Seeingthat there was an un-roundness of something like a millimeter, I decided to put thewhole thing in the lathe again. After a lot of spray puttying and grinding it was backto the lacquering box. Then of course all the air holes in the glass fibre surfaced andI had some nice sessions of applying spray putty and grinding it in the lathe again.Then it was back to the lacquering box. To avoid patches becoming visible whenrotating the spacetime, I hooked it up to an electric drill rotating the spacetime asI just moved the paint sprayer from one end to the other. A few more things wentwrong.

• Putting a finger in the un-dried paint to see if the paint isnot dry yet -- a classic!

• The spacetime spinning to fast - paint drying to fast.

• Un-smoothness of the surface from the last lathing becoming vi-sible as the metallic is applied.

In the end I made more than 15 lacquering attempts – and every time you haveto wait for the paint to dry, grind the spacetime, flush the lacquer box and yourself(wearing a rain coat) etc. I even left it to a professional car lacquerer at one point, butI was not happy with what he had done either, so I ground it again and kept going.Suffice it to say that in the end I made it.Oh – just in case you want to try this – some inspectors found out about my

technique with the electric drill and they were not overly happy to have a spark-inducing machine inside a flammable fog of thinner:)Putting the coordinate lines on the surface was analogue to what I did on the

funnel. Parallel to making the bulgy spacetime, I also made a flat spacetime model,using a plane of PVC-plastic. My models are displayed in Fig. 4.3.

4.2. The bulgy cylinder spacetime 25

Figure 4.3: Some curved shapes on my bed:)

26 4. Making models

5Spacetime visualization

I have written two research papers where I develop two different techniques directedtowards popularizing the field of general relativity. Both these techniques allows usto visualize curved spacetime by a curved surface. The ideas underlying the illus-trations are however quite different, and they illustrate different aspects of the fulltheory of General Relativity.Similar techniques have been developed before. There is a popular scientific book

called ’Visualizing Relativity’, by L. C. Epstein [1]. Some of the illustrations are quitesimilar to the ones that I have created (related to Paper I) although the underlyingidea and how one interprets the illustrations is completely different1.Yet another technique is presented by D. Marolf in [2] though interpreting the

embedded surface that he considers requires some knowledge of special relativity 2.There is also a paper by W. Rindler [3] where the basic ideas have some similarity

to those of Paper I though he considers a more mathematical approach intended forundergraduate students of physics. 3.

1The theory underlying [1] is based on the assumption of an original time-independent, diagonal,Lorentzian line element. Rearranging terms in this line element one can get something that looks likea new line element, but where the proper time is now a coordinate. The ’space-propertime’ can beembedded as a curved surface.

2In particular he considers the radial line element of a maximally extended black hole. The properdistances can be illustrated by embedding the surface in 2+1 dimensional Minkowski space (visual-ized as a Euclidean 3-space)

3Essentially he considers the Newtonian equations of motion in Hamilton’s formalism, andmatches them (approximately) with the geodesic equations of motion we get considering a four-dimensional Riemannian (positive definite) line element. The components of the metric are foundusing qualitative guessing and trial and error. Thus he presents a way of bypassing special relativ-ity to anyway give an understanding of the concept of curved spacetime – given some knowledge ofNewtonian mechanics and Riemannian geometry.

27

28 5. Spacetime visualization

5.1 Paper IIn this paper I present a method that allows us to visualize curved spacetime by acurved surface. The method is tailor-made to explain what it means to have straightlines in a curved spacetime and how this can explain why an apple thrown upwardsreturns to Earth. This method is underlying the introduction to general relativitypresented in chapter 2, as well as the cover of this thesis.The idea underlying this method is hard to explain in brief without getting tech-

nical, but I will try to give a feeling for what is done in the paper.Consider two events, like snapping your fingers, first with the right hand and

then with the left. In Einstein’s theory, one assigns an interval (a kind of distance)between this pair of events. If there is time for a light signal to travel from one eventto the other, the interval will be positive otherwise it will be negative 4.Next consider the curved surface of a sphere onwhichwe have sprinkled grains of

sand. The geometry of the sphere can be defined as the set of distances separating allpairs of (nearby) grains of sand on the surface. Similarly, the geometry of spacetimecan be defined as the set of intervals separating all pairs of (nearby) events. Relativeto this abstract geometry, a canon ball shot from a canon will follow a line that isstraight in some (abstract) sense. The fact that we have negative intervals (distances)between events however makes it impossible to illustrate directly the spacetime ge-ometry by a sphere or some other curved surface.What I do in Paper I is that I take the set of spacetime intervals between events,

and transform them such that all the intervals become positive, but without changingwhat is a straight line. We can illustrate the new geometry, that has only positivedistances, with a curved surface.

5.1.1 A technical noteIn technical terms, I consider a Lorentzian two-dimensional, time-independent lineelement. I then find another line element that is positive definite but has the samegeodesic structure. The new geometry can be embedded in Euclidean space. Dueto a one-parameter freedom in going from the original to the new dual line element,together with freedoms of the embedding, one can illustrate even the weakly curvedspacetime of Earth with a significantly curved surface.

5.2 Paper IIIn this paper I present another method of visualizing curved spacetime. This methodis well suited to explain why clocks slow down near massive bodies, and what that

4Strictly speaking it is the square of the interval that can be positive or negative – but that is of littleimportance here.

5.2. Paper II 29

really means. The method can also be applied to explain black holes, expanding uni-verses (big bang) and much more. This is a technical paper exploring the possibilitiesof the method rather than applying the method. However, in section 2 of the paperthe method is applied to give a very brief introduction to general relativity. The sec-tion is directed to teachers of physics, and uses only a little bit of mathematics, thoughin principle the method can be used without any reference to mathematics. Even atthe current level, a general reader with an interest to learn about curved spacetimemay benefit from reading it. Fig. 5.1 gives an example of an illustration from PaperII.

Space

Time

Figure 5.1: Cosmological models. From thesemodels onemay understand how spaceitself can expand. For further details, see section 2 of Paper II.

Aswas the case in the previous paper, the idea is tomake a transformation turningall of the intervals (distances) between pairs of events into positive distances. Thistime the idea is however not to preserve straight lines, but to preserve the intervalsthemselves (as much as possible). The resulting geometry can be illustrated with acurved surface. If one knows how to interpret such an illustration, one will be ableto find out everything there is to know about the true spacetime geometry (includingthe negative distances).

30 5. Spacetime visualization

5.2.1 A technical noteTechnically speaking, I first introduce an arbitrary field of timelike four-velocities uµ.Then, at every point, I perform a coordinate transformation to a local Minkowskisystem comoving with the given four-velocity. In the local system, the sign of thespatial part of the metric is flipped to create a new metric of Euclidean signaturewhich for the special case of two dimensions be embedded as a curved surface. Onthe surface lives small Minkowski systems relative to which special relativity holds.

6Inertial forces

An example of an inertial force is the (apparent) force that pushes us outwards if weare on a rotating platform, or if we drive our car through a roundabout at high speed.This particular inertial force is known as a centrifugal force.In the general theory of relativity one can introduce inertial forces in a similar

manner to how one introduces them in Newtonian mechanics. Some effects that oc-cur in relativity are however quite counter-intuitive from the point of view of Newto-nian mechanics. In fact if we consider a rocket in orbit near a black hole it will requirea greater rocket thrust outwards to keep it from falling into the black hole the fasterit rotates around the black hole1. We might say that the centrifugal force is pointinginwards rather than outwards here.

Earth

Figure 6.1: Left: a rocket orbiting the Earth. The faster the orbital velocity the lessoutward thrust from the rocket engine is required to keep the rocket on a fix radius.If the orbital velocity is high enough – no outward thrust is required. Right: a rocketin orbit around a black hole. The faster the rocket moves the greater the requiredrocket thrust needed to keep it from falling into the black hole.

1This effect occurs between the event horizon (the radius of the black hole) and 1.5 times the radiusof the black hole (the so called photon radius where particles of light can move on circles).

31

32 6. Inertial forces

I consider inertial forces in general spacetimes using the mathematical formalismof general relativity in Paper III. I also derive a general formalism of inertial forces us-ing only basic principles of relativity as presented in paper IV. The interested readerwho knows a bit about special relativity and vectors may benefit from reading thispaper. Among other things, I show how one can explain the abovementioned sce-nario with the black hole as a natural consequence of relativity. The mathematicalresults of the formal paper are a bit more general than those of the intuitive paper,but the results agree where comparable.

6.1 A technical noteIn Paper III we consider a general timelike congruence of reference worldlines in anarbitrary spacetime. The local motion of a test particle can be described in terms ofthe velocity relative to the congruence, the time derivative of this velocity and thespatial curvature of the test particle worldline projected onto the local slice. In thisformalism inertial forces appear naturally in form of the kinematical invariants of thecongruence (see chapter 12). While relativistically correct, the resulting equations ofmotion are effectively three-dimensional. I show that when the congruence is shear-ing, the projected curvature is not necessarily the most natural measure of spatialcurvature, and I present an alternative definition. I also study the effect of conformalrescalings on the inertial force formalism.In Paper IV, via the equivalence principle and basic elements of special relativity

such as time dilation, I derive the same formalism using only three-vectors for thespecial case of a non-shearing congruence.

7Gyroscope precession

A gyroscope is basically a symmetric body that spins very fast around an axis andis suspended in such a way that there is no torque acting on it. If we move the gy-roscope along a circle it will keep pointing in the same direction according to thetheory of Newtonian mechanics. According to special relativity however, if we trans-port the gyroscope very fast along the circle – its spin axis will turn, or in other wordsprecess. As we go to general relativity the situation becomes even more interesting.For instance, if we move the gyroscope along a certain circle around a black hole –the gyroscope spin axis will (automatically) turn in such a manner that the spin axisis always directed along the direction of motion (see Fig. 7.1).

Figure 7.1: A gyroscope moving on a circle around a black hole. The spin axis is thethick bar through the sphere. Despite the fact that we are not affecting the gyroscopeby any torque, it still turns.

33

34 7. Gyroscope precession

I consider gyroscope precession in general spacetimes in Paper V using the math-ematical formalism of general relativity. I also consider gyroscope precession in avery non-formal manner in Paper VI. Here I use basic principles of relativity and in-tuition rather than formal mathematics. In particular I show how one can use thespecial relativistic explanation to explain also the general relativistic effects. The for-mal paper is a bit more general than the intuitive paper but the results match wherecomparable.

7.1 A technical noteAswas the case in the inertial force analysis, the idea rests on introducing an arbitrarycongruence of timelike worldlines and expressing the spin precession with respectto this frame of reference. Rather than considering the standard spin vector Sµ, Iconsider the spin vector we would get if we would stop the gyroscope by a pure boostrelative to the congruence. The stopped spin vector obeys simple laws of rotationand is ideally suited for this approach. In Paper V I use a four-covariant formalismstarting from the Fermi-Walker transport equation, and derive an effectively three-dimensional formalism of spin precession.In the intuitive paper I arrive at the same result, considering a non-shearing refer-

ence frame (congruence) using only three-vector formalism together with the equiv-alence principle and special relativity. I do not use the Fermi Walker transport equa-tion in this paper. In particular I show that one may regard the gyroscope precessionin arbitrary stationary spacetimes as a double Thomas precession effect. One partcomes from the gyroscope acceleration and the other from the reference frame accel-eration (there is also a trivial contribution from any rotation of the reference frame).

8Optical geometry

The mass of a star curves the fabric of space and time. How space is curved is illus-trated in Fig. 8.1.

Figure 8.1: Illustrating the curved geometry of a plane through a star.

While the velocity of light is everywhere locally the same, clocks near the starwill run slow relative to clocks far from the star (for some understanding of this, seesection 2 of Paper II). Effectively this means that as we send a light signal from oneend of the star to the other (imagine that the star is transparent), it will take a longertime for the light signal than we might think considering only the spatial distancethat the light needs to travel. Seen from the outside, there is an apparent slowingdown of the light signal. We can account for this by considering a rescaled space,where the extra distance accounts for why the light takes such a long time to reachthe other side of the star. This stretched space is known as the optical geometry, seeFig. 8.2. Relative to the optical geometry, photons (particles of light) move alongstraight spatial lines. This is not generally the case for photons. If we send out aphoton horizontally – it will fall away from a straight line just like anything elsewe might throw in a horizontal direction. The only difference is that photons moveso fast that we do not see them fall. But relative to the optical geometry – they dofollow straight lines. In [16] M. Abramowicz gives a popular scientific presentationof optical geometry.Together with my collaborator Hans Westman, I present in paper VII a way of

35

36 8. Optical geometry

Figure 8.2: The optical geometry of a plane through a star. It is related to the standardgeometry of the plane by a stretching. The darker region lies within the star and thelowest point is the center of the star. Unlike the curved surfaces presented prior tothis chapter, this is a visualization of curved space and not of curved spacetime.

generalizing the standard theory of optical geometry to include arbitrary spacetimes(so that the optical geometry may change in time). In paper VIII I consider specificapplications of the generalized optical geometry. In particular I consider a black hole,including its interior.

8.1 Generalized optical geometry in technical termsIn an arbitrary spacetime we may introduce a spacelike foliation specified by a singlefunction t(xµ). Forming the covariant derivative of this function we get a vector fieldorthogonal to the foliation. We then introduce a congruence of worldlines parallel tothe vector field. Performing a conformal transformation we rescale away time dila-tion, so that in coordinates adapted to the slices and the congruence we have gtt = 1(here the tilde indicates a rescaled object). The new spatial geometry that we get is theoptical geometry. Relative to this geometry a photon moves a unit distance per unitcoordinate time. Assuming the congruence to be shearless we show that a a geodesicphoton will follow a spatially straight line, and a gyroscope following a straight spa-tial line will not precess relative to the direction of motion. Also the sideways forcerequired to keep a test particle moving on a straight spatial line will be independenton the velocity. Using a novel measure of spatial curvature introduced in Paper IIIwe show that also for the case of a shearing congruence, photons follow straight linesand the sideways force keeping a test particle moving along a straight line is inde-pendent on the velocity. In paper VIII I consider a non-shearing congruence radiallyfalling towards a black hole, allowing us to consider the extended optical geometryacross the horizon.

9Conclusion and outlook

Concerning my work with inertial forces, gyroscope precession and optical geometry– I have largely accomplished what I set out to do. At the moment, I see no particularissues here that I would like to further resolve.Concerning the spacetime visualization techniques, I ammore or less content con-

cerning the theoretical background. I am however a little curious to find out what canbe done in the dual scheme considering time dependent line elements. Also I am bitcurious about for instance how the spacetime of a radial line through a collapsingshell of matter would look like in the absolute scheme. Mainly however, my interestconcerning the pedagogical techniques lies in the visualization as such. I have plentyof ideas of how one can go beyond the illustrations that I have shown in this thesis tofurther inspire and teach the physics community and the general community aboutthe marvels of curved spacetime. As I mentioned in the introduction, I hope to usemy ideas and methods to write a book about relativity for a general audience.As for my interest in physics I tend towards the foundations of physics. How can

one unite general relativity with quantum mechanics? Maybe if we crack this nutwe will get a grip on the measurement problem of quantum mechanics, and perhapseven consciousness itself. Or maybe it will be the other way around. In any case, Isuspect that a revolution in the way we think about the universe will be necessary toaccomplish this fusion. As soon as I have finished this thesis – I will get started on it.It shouldn’t take me very long to crack that nut:)

37

38 9. Conclusion and outlook

Comments on theResearch Papers

10A spherical interior dual metric

This chapter is a comment on Paper I, concerning the shape of spacetime for a radialline inside a planet of constant proper density. It presumes knowledge of the notationof Paper I.We know that in Newtonian theory, a particle that is in free fall around the center

of a spherical object of constant density, will oscillate with a frequency that is inde-pendent of the amplitude. This would fit well with an internal dual space that isspherical. We saw in Paper I the possibility to choose parameters ! and k that pro-duces substantial curvature also in the case of the weak gravity outside our Earth. Itis natural to ask whether we really can find parameters such that the internal geome-try becomes spherical. In the following sections we will see that this is not generallythe case but one can choose parameters such that it is exactly spherical in the weakfield limit.

10.1 Conditions for spheres

For a sphere of radius R, we introduce definitions of r and z according to Fig. 10.1.Using the Pythagorean theorem it follows that:

dr

dz= #

!"R

r

#2

# 1 (10.1)

We let z = z(x), where (see Paper I, Eq. (23)):

dz

dx=

$c(x) # r!2 here r! =

dr

dxc = gxx (10.2)

41

42 10. A spherical interior dual metric

Rr

z

Figure 10.1: Definitions of variables for a sphere. The axis of rotation (around whichtime is directed) is horizontal.

We readily find:

R2 =r2

1 # r!2/c(10.3)

Also if we have a rolled-up sphere (see Fig. 6 of Paper I) such that r = "rsphere andr! = "r!sphere this is modified to:

R2 =r2

"2 # r!2/c(10.4)

10.2 The dual interior metricFor the Schwarzschild interior dual metric we may show that 1 :

r!2/c =!k2x2

4x60(a0 # !)

and r2 = #k2 a0

a0 # !(10.5)

Inserting this into Eq. (10.4) we readily find that for the embedding to correspond toa sphere of radius R and rolling parameter " we must have, for all x < x0:

R2"2(a0 # !) # R2!k2x2

4x60

# #k2a0 = 0 (10.6)

Notice that if we divide the expression by #k2 we will see that any change in R2 and" can be canceled by a corresponding change in # and k respectively.For Eq. (10.6) to be true for all x it must be true for every power in x. We note that

a0 has a lot of powers in x which would mean that the factors multiplying a0 wouldhave to vanish. But then the x2-term in the middle cannot be canceled by anything.Thus the expression cannot be true for all values of x and thus the interior dual metricis not exactly spherical.

1The first relation can most rapidly be obtained re-juggling Eq. (32) in Paper I (replace < with =and remove the “min”), look also at Eq. (23) for understanding. The second is the square of Eq. (21)in Paper I.

10.3. Approximative internal sphere 43

10.3 Approximative internal sphereWe can however produce something that is very similar to a sphere. Set " = 1, corre-sponding to a non-rolled sphere, and # = (1 # !)/k2 meaning unit radius at infinity.We can expand the two occurrences of a0 in Eq. (10.6) to second order in x2. De-manding the equation to hold to zeroth, first2 and second order yields after somesimplification (Mathematica does fine simplifications):

! =a00 · (R2 # 1)

R2 # a00where a00 $ a0(0) =

1

4

%

3

!

1 # 1

x0# 1

&2

(10.7)

k =2x7/4

0$3%

x0 # 1 #%x0

(10.8)

Notice that Eq. (10.8), coming directly from the second order demand, is independentof !. The first relation is nothing but Eq. (21) in Paper I, where one has demandedr = R and # = (1 # !)/k2, taken at x = 0.Inserting ! and k from Eq. (10.7) and Eq. (10.8) into a plotting program yields

pictures with a very spherical appearance. The way it works is that the center ofthe bulge has the exact radius and curvature of a sphere, then the rest is not exactlyspherical. In the Newtonian limit however, where x0 & ', we do get a perfect sphere.This is true whichever finite radius Rwe choose, as is explained below.

10.4 Spheres in the Newtonian limitFrom Eq. (10.3) we have the necessary relation for a sphere:

r!2

c= 1 # r2

R2(10.9)

Using the specific expressions for ! and k given by Eq. (10.7) and Eq. (10.8) we mayevaluate both the left and the right hand side of Eq. (10.9) to lowest non-zero order 3in 1/x0, to find that the equality holds exactly for all x to this order. Thus in this limitwe get an exact sphere. In fact we knew in advance that the equality would hold.The reason is that, a0 expanded to second order in x for which the equality holds, isthe same (or in fact a bit more exact) as a0 expanded to first order in 1/x0

4. So, in theNewtonian limit the interior dual metric is isometric to a sphere for any value of !(so long as R remains finite).

2It is always satisfied to first order3Terms like x2/x3

0 are of course treated as 1/x0-terms4If we instead would have had a0 = 1 #

'1 # x3/x4

0 which expanded in x to second order is zero,while not being zero expanded to first order in 1/x0 (remember that x3/x4

0 is of order 1/x0), we couldnot have used this little trick.

44 10. A spherical interior dual metric

An embedding of the Earth spacetime is displayed in Fig. 10.2. Here I have chosenthe radius of the central bulge to be twice the embedding radius at infinity. Theparameters ! and k are given by Eq. (10.7) and Eq. (10.8) with x0 = 7.19 · 108, suitablefor the Earth. As can be seen the bulge is quite spherical.

Figure 10.2: An embedding of the dual spacetime of a central line through the Earth.The time per circumference is roughly 84 minutes.

Notice the worldlines of the two freefallers on the interior sphere – one static inthe center – the other oscillating around the Earth with a period time of roughly 84minutes.

11The absolute visualization

In this chapter I take the opportunity to comment on some issues concerning theabsolute approach to spacetime visualization (Paper II). I will consider only two-dimensional applications.I start by giving some intuitive understanding for how changing generators af-

fects the visualization. I then move on to find what the vacuum field equations looklike in 2 dimensions assuming a flat absolute geometry (so the freedom lies in thetwist of the generators).I will also consider some toy-models of theoretical interest, in particular illus-

trating spacetimes with timelike loops. I will also give an example of how onemay choose generators such that outgoing photons (for instance) follow absolutegeodesics for the line element of a radial line through a black hole.

11.1 The perch skin intuitionIn special relativity one can look at the active Lorentz transformation as a shift of thephysical spacetime points along hyperbolas, as illustrated to the left of Fig 15 in PaperII. Alternatively we may view the Lorentz transformation as a two-step stretch andcompress process along the light cone coordinates u and v. Stretching by a certainfactor along umeans compressing by the same factor along v (so areas are preserved)as depicted in Fig. 11.1.This process of stretching and compressing is directly related to the absolute vi-

sualization scheme. If we choose generators in the leftmost rhomboidal section ofspacetime in Fig. 11.1, such that the generators are rotated a certain angle clockwisefrom the upwards direction – then the absolute geometry would be that of the surfaceto the right (think of the generators in the comoving coordinates). Hence changingthe generator corresponds to a compression and a stretch along the local lightcone

45

46 11. The absolute visualization

related toAnd this is

Stretch

u v

Compressmy skin?

Figure 11.1: An active Lorentz transformation corresponds to a stretch and a com-pression.

coordinates. This reminds me of how a perch skin is behaving. Pulling in the skinslength-direction makes the skin automatically shrink in stripe direction 1.We realize that for any embedded surface in the absolute visualization technique,

we can locally stretch and compress it to give a different appearance to the same phys-ical spacetime. In other words we can do any deformation that preserves local areasand keeps the null lines at right angles. As an application of this newfound intuition,we consider a deformation of a Lorentzian flat spacetime illustrated by a flat absolutemetric with uniformly directed generators – to a new flat absolute metric where thegenerators are curving, see Fig. 11.2.

Stretch

Stretch

Press

Press

u v

Figure 11.2: A deformation of flat spacetime. All the lines are null except the thickcurve which is a generating worldline. Any deformation that preserves the anglesbetween null lines and preserves local areas will leave the Lorentzian spacetime un-affected.

1As soon as my father starts hooking all those perches that he claims are biting his lure, I can startmaking more extensive tests of the properties of perch skin:)

11.2. Vacuum field equations for flat absolute metric 47

11.2 Vacuum field equations for flat absolute metricIn the absolute visualization scheme we have have a direct visual means of findingthe proper distances between nearby events. Maybe we could also build some intu-ition of how to find out whether there is a proper curvature or not. For this purpose,we study a simple case of flat absolute geometry where the Lorentzian geometry isdetermined by the direction of the generators, specified by an angle # as depicted inFig. 11.3.

t

x

!

Figure 11.3: The direction of the generators (the local Minkowski systems) specifiedby an angle #.

The angle #(t, x) goes clockwise from the t-axis on the plane to the local t-axisof the local Minkowski system. The absolute four-velocity of the generators has theform:

uµ = (cos(#), sin(#)) (11.1)We find the Lorentzian metric through gµ! = #$µ! + 2uµu! as:

gµ! :

(

))*#1 + 2 cos2(#) 2 cos(#) sin(#)

2 cos(#) sin(#) #1 + 2 sin2(#)

+

,,- (11.2)

We may calculate the Ricci tensor for this using grtensor in Mathematica, but theexpressions become quite large. To simplify matters, we choose our t-axis to coincidewith the local Minkowski time axis. Then the angle # is small in a region around thispoint and we may Taylor expand the metric (to second order in #):

gµ! :

(

))*1 # 2#2 2#

2# #1 + 2#2

+

,,- (11.3)

Calculating Rµ! = 0 and setting # = 0 with grtensor yields a single equation:(%x#)2 # (%t#)2 + %x%t# = 0 (11.4)


So here we have the vacuum field equations 2 , assuming a flat absolute geometry.Notice that Eq. (11.4) holds in a point, where the t-axis is chosen to coincide withthe generator. Notice also that the two first terms are first order derivatives whereasthe last is a second order derivative. At any point we can thus choose an arbitraryderivative in the local t and x-direction and still have a locally flat spacetime – so longas %x%t# is given by Eq. (11.4).

11.3 On geodesics and flat metricsAs an application of the field equations above, consider the special case of a flat abso-lute metric in 2D, and geodesic generators (i.e straight lines). Notice that the genera-tors do not necessarily have to be parallel, but could for instance extend from a pointand outwards (like spokes on an old cartwheel), or something more complicated.Assuming the generators to correspond to straight lines, we have everywhere:

(# · n = 0 where n = (cos(#), sin(#)) (11.5)

Here # is the angle describing the tilt of the local generator as depicted in Fig. 11.3. InCartesian coordinates (t, x) where the t-axis is parallel to the generators at the pointin question, we have # = 0 and thus Eq. (11.5) yields %t# = 0. Differentiating Eq.(11.5) with respect to x, and then setting # = 0 yields:

(%x#)2 + %x%t# = 0 (11.6)

Comparing with Eq. (11.4) we see that the vacuum field equations are satisfied. Thusfor geodesic generators on the flat plane the corresponding Lorentz geometry is flat.For an arbitrary embedded spacetime in the absolute visualization scheme – con-

sider turning the lightcone (the local generator) by 90" everywhere while letting theshape of the surface remain. This would exactly correspond to changing the signof the Lorentz metric. Changing the sign of the metric does not affect the affineconnection, and thus not the curvature either. It then follows that also if we havestraight lines orthogonal to the generators for a flat absolute geometry – we have a flatLorentzian geometry 3.Indeed looking at the rightmost figure in Fig 15 of Paper II we have a good exam-

ple of a flat absolute metric (embedded as a cone) with straight generators in someparts, and straight orthogonal lines in another part, that is Lorentzian flat.

2In two dimensions the vacuum field equations imply flat Lorentzian spacetime.3We could also show this directly assuming that we everywhere have (! ·.

cos(! + !2 ), (sin(! + !

2 )/

= 0. It follows analogously to the above derivation (differentiate withrespect to x this time) that also for this case the vacuum equations are automatically satisfied, hencespacetime is flat.

11.4. Closed dimensions and timelike loops 49

11.4 Closed dimensions and timelike loopsThe absolute illustration method is well suited for making spacetime toy models.Such illustrations can be valuable for thought experiments and qualitative reasoning.In this section we illustrate a few spacetimes that are closed in one or two dimensions.In particular, images as that depicted to the left in Fig. 11.5 are useful to illustrate howspacetime can be something much more complex than just space plus time.

Figure 11.4: To the left a spacetime that is closed in space, with the trajectories of twoinertial observers. To the right a spacetime that is closed in time.

Figure 11.5: To the left a spacetime that is closed in space in some regions (top andbottom) and closed in time in the middle region. To the right a spacetime that isclosed in space and in time.


An observer whose worldline winds as depicted to the left in Fig. 11.5 would asyoung see himself as old and vice versa. The thought that easily comes to mind is –’What if he as young kills himself as old?’ or perhaps worse still ’What if he as oldkills himself as young?’. Obviously this raises rather interesting questions concerningfree will. I will however leave it to the reader to further consider this.

11.5 Warp driveIn science fiction terms like warp drive have become standard as a means of trans-portation that is in some sense faster than light. One might ask to what extent such atransportation is possible within the framework of general relativity. Well, if we havesome way of preparing spacetime ’from outside the spacetime’ this is certainly not aproblem as depicted in Fig. 11.6.

Figure 11.6: A warp drive spacetime.

As always – for any spacetime that we can come up with – there is an energy-momentum tensor that via the field equations gives the right local curvature andwillautomatically obey the local conservation laws of energy and momentum (via theBianchi identities).Assume now that in our lives everything operates as if we are living through a flat

spacetime. Assume further that we have some means of freely manipulating localmomentum and energy currents (and that energy may be negative etc). Then if weplan ahead we can send out emissaries that will tweak the energy momentum tensorat some preset time (according to their own clocks) and thus create from within thespacetime a warp bridge like that depicted to the right in Fig. 11.6 or like Fig 17 of

11.6. Finding generators to make a single trajectory an absolute geodesic 51

Paper II. Of course this method of transportaion is no faster than if you would jointhe rightmost emissary and just go there. It is different however. In the warp methodyou create an opportunity to travel fast should you so choose at a later time.Of course we have assumed quite much here, the least of which is the ability to

make a global smooth tweak of the energy momentum tensor – which is not practi-cally feasible. We could perhaps hope to make a series of discrete explosions or somesuch. However, in the classical theory I see no problems in principle of the abovetype of warp drive.Of course if we consider quantum mechanics – it is not at all clear what we can

and cannot do for this case. Indeed we would need a theory of quantum gravity toexploit this fully.We could consider a corresponding emissary scenario also for the case of an ini-

tially flat spacetime with closed space – to at will create a spacetime as that depictedto the right of Fig. 11.5. Again I will leave it to the reader to consider whateverparadoxes concerning free will that might occur there.

11.6 Finding generators to make a single trajectory anabsolute geodesic

This section is essentially an appendix that for brevity was cut out from Paper II. Thenotation is therefore directly related to that of Paper II and references to equations inPaper II will be of the form ’Eq. II.xx’.In this section we assume both the original and the absolute metric to be 2-dimen-

sional, time independent and diagonal. For a geodesic relative to the absolute metricwe have (in the coordinates for which it is diagonal):

%dx!

dt!

&2

=g!

tt

g!xx

(&!g!tt # 1) (11.7)

Here &! is a constant that is fixed for every geodesic. Suppose now that we wouldwant somemotion in the original metric, to be a geodesic in the absolute metric. It willthen turn out to be practical to express the original motion using the Killing velocity,denoted by w for the motion considered. The corresponding four-velocity, prior todiagonalization, is given analogous to Eq. II.14 as:

qµ = ±!

gtt

1 # w2

%1

gtt,

#w%#gxxgtt

&

(11.8)

Using Eq. II.25, and Eq. II.26 we can transform qµ to q!µ. Then we can express dx!

dt! asq!x

q!t which is then given as a function of w. Using this is in Eq. (11.7), together withEq. II.15 we find after simplification:

w2(1 # v2)2

(1 + v2 # 2vw)2#

%

&!gtt1 + v2

1 # v2# 1

&

= 0 (11.9)


The generating velocity v has to obey this equation to make an original motion, char-acterized by w(x), a geodesic in the absolute metric. In general it would be tricky tosolve this equation for v. We can factorize it, but we still get a fourth order equationin v. There are however cases where we can deal with it analytically.

11.6.1 Considering generatorsSetting v = w, thus demanding the generators to be absolute geodesics, Eq. (11.9) isreduced to:

v2 = 1 # &!gtt (11.10)

It is easy to show that this corresponds exactly to a geodesic in the original space-time. Thus we draw the conclusion that the generators of the absolute metric will begeodesics in the absolute spacetime if and only if they are geodesics in the originalspacetime. While we here arrived at this conclusion assuming a Killing symmetryamong other things, it is a completely general result as is shown in Appendix II.E.

11.6.2 Considering photonsAnother case where Eq. (11.9) is trivialized is when we consider an outward-movingphoton (w = #1). Then the solution is:

v =2 # gtt&!

2 + gtt&! (11.11)

Here &! is an arbitrary constant. Assuming a Schwarzschild line element and choos-ing v = 0 at infinity yields &! = 2. For this particular example we may insert the v ofEq. (11.11), and the line element of Eq. II.10, into Eq. II.15 to find:

g!µ! =

0

1112

12

31 + (1 # 1

x)24

0

0 21+(1# 1

x )2

5

6667 (11.12)

We see that the absolute metric exists all the way into the singularity. We may how-ever notice that the Killing velocity v, as defined by Eq. (11.11), becomes infinite atx = 1/2. That is however nothing to worry about. It just means that at this point thegenerators are exactly orthogonal to the Killing field. In Fig. 11.7 we see an embed-ding of the geometry described by Eq. (11.12).The thick lines with arrows correspond to photons moving outwards. The left one,

being inside the horizon, is however guided by the geometry into the singularity 4.4Strictly speaking the photon trajectories should be invisible after having spiraled once around the

surface since the spacetime in this embedding is layered.

11.6. Finding generators to make a single trajectory an absolute geodesic 53

Figure 11.7: The absolute spacetime using generators uµ(x) such that outward-moving photons follow geodesics. The radial parameter x lies in the interval [0.55, 4].

The horizon is located where there is a minimum in the embedding radius. This isnecessary if we want the photon to remain on a certain spatial position and at thesame time follow a geodesic on the rotational surface. Also it has to be a minimumor a photon could oscillate back and forth around the horizon.While outward-moving photons correspond to geodesics, it is apparent that in-

ward-moving photons do not. To see this we consider an inward-moving photon atthe horizon. The trajectory is there directed purely along the surface (no azimuthalvariation). A geodesic tangent to the photon trajectory at this point will remain ona fix azimuthal angle, whereas the photon trajectory does not. Thus, without us-ing any mathematics, we may understand that we cannot make both ingoing andoutgoing photons correspond to geodesics, while keeping manifest time symmetry(and assuming that we want to embed the horizon). By relaxing the time indepen-dency one could hope not only to get inward- and outward-moving photons to followgeodesics, but all free particles. In Appendix II.D, II.F and II.G, we show that this ispossible only in a very limited class of spacetimes.


12Kinematical invariants

In Papers III-VIII we employ a congruence of reference worldlines that threads thespacetime1. The local behavior of the congruence can be described by the covariantderivative of the congruence four-velocity. This derivative can be split into differ-ent parts, known as the kinematical invariants of the congruence. They are tensorsrelated to shear, expansion, rotation and acceleration. In this chapter we give someintuition regarding the meaning of these invariants and their relation to the covariantderivative of the congruence four-velocities.

12.1 The definitions of the kinematical invariantsThe kinematical invariants of a congruence of worldlines of four-velocity uµ are de-fined as (see e.g [7] p. 566):

aµ = u"("uµ (12.1)' = ("u" (12.2)

&µ! =1

2(P #

!(#uµ + P #µ(#u!) #

1

3'Pµ! (12.3)

(µ! =1

2(P #

!(#uµ # P #µ(#u!) (12.4)

In order of appearance these objects are denoted the acceleration vector, the expan-sion scalar, the shear tensor and the rotation tensor. The projection operator Pµ

" isgiven by (using the (#, +, +, +) convention):

P µ" = gµ

" + uµu" (12.5)1In the more intuitive Papers IV and VI, we do not explicitly talk of reference worldlines threading

spacetime, but effectively they are there anyway and this chapter is relevant also for these papers.

55

56 12. Kinematical invariants

Forming Pµ"k" yields a vector corresponding to the part of kµ that is orthogonal

to the congruence (as is obvious in inertial coordinates locally comoving with thecongruence).We may also introduce what we denote as the expansion-shear tensor:

'µ! =1

2((#uµP

#! + (#u!P

#µ) (12.6)

From the normalization of uµ it follows that u"(µu" = 0. Also we know that thecovariant derivative of the metric vanishes. Then we can write Eq. (12.4) and Eq.(12.6) in a slightly different but equivalent form:

(µ! =1

2P #

µP$

!(($u# #(#u$) (12.7)

'µ! =1

2P #

µP$

!(($u# + (#u$) (12.8)

In this form the three-dimensional nature of the rotation tensor and the expansion-shear tensor is more obvious. From the definitions it follows that:

(!uµ = (µ! + 'µ! # aµu! (12.9)

So here are the definitions of the kinematical invariants. Now let us see if we canunderstand the physical meaning of these objects. In particular we will focus on therotation and the expansion-shear tensor.

12.2 The average rotationThe rotation tensor is evidently (by its name) connected to rotation of the congruencepoints. The tensor is however well defined also when the congruence is shearing anddeforming, in which case there is no rigid rotation. We may suspect that for this casethe rotation tensor is related to an average rotation of the reference congruence points.With this in mind let us derive an expression for the average rotation, and see if theresulting expression is connected to the tensor (µ! .Consider then a collection of moving reference points in three-dimensional Eu-

clidean geometry. Also assume that the velocity at the origin of the coordinates inquestion is zero momentarily. We start by deriving an expression for the averagerotation around the z-axis, considering a circle of radius r in the z = 0 plane. Theaverage rotation of the points along the circle is r#1 times the average velocity in thecounter-clockwise direction. Letting u(x, y, z) be the velocity of the reference points,we may express the average rotation as a line integral over the circle:

(z =1

r

1

2)r

8

"

u ) dx =1

r

1

2)r

8

•

((" u) · dS (12.10)

12.2. The average rotation 57

In the last equality we have used Stokes theorem, turning the line integral over thecircle into a surface integral over a circular flat disc with dS = zdS. In the limit asthe radius of the circle goes to zero,("u can be considered as constant (to the ordernecessary) and we can move it out of the integral:

(z *1

r

1

2)r((" u) · z

8dS =

1

2((" u) · z (12.11)

So in the limit as the radius of the circle goes to zero we have:

(z =1

2(%xuy # %yux) (12.12)

This is then the average rotation around the z-axis for the motion of the referencepoints in the z = 0 plane (in the limit where the circle over which we average goes tozero). We realize that also averaging over the the rotation considering non-zero z, Eq.(12.12) gives the local average rotation around the z-axis. Corresponding argumentsgives us the average rotation around the other axes:

! =1

2(%yuz # %zuy, %zux # %xuz, %xuy # %yux) (12.13)

So here we have the average rotation vector that we set out to find. For some furtherintuition see Fig. 12.1.

y

x

Figure 12.1: A shearing velocity field in two dimensions. The reference points alongthe x-axis are rotating counter-clockwise whereas the points along the y-axis havea clockwise rotation. The average of the corresponding angular velocities (countedpositive in the counter-clockwise direction) is (z, which for this case is zero.

12.2.1 A matrix formulation of rotationForming ! " x we get the contribution coming from an (average) rotation to thevelocity at a point x :

! " x = ((yz # (zy, (zx # (xz, (xy # (yx) (12.14)


Now form a matrix (ij :

(ij =1

2(%jui # %iuj) (12.15)

In matrix form this becomes:

(ij :1

2

(

))))*

0 %yux # %xuy %zux # %xuz

%xuy # %yux 0 %zuy # %yuz

%xuz # %zux %yuz # %zuy 0

+

,,,,-=

(

))))*

0 #(z (y

(z 0 #(x

#(y (x 0

+

,,,,-(12.16)

It follows that we have:

(ijx

j = ((yz # (zy,(zy # (yz,(xy # (yx) (12.17)

The right hand side of this equation is identical to the right hand side of Eq. (12.14).So we understand that (i

jxj , where (ij is defined by Eq. (12.15), gives the velocity of

the congruence point at xi coming from an average rotation.It is not hard to realize that we can form the rotation tensor as defined by Eq.

(12.15) from the rotation vector as 2:

(ij = #*ijk(k (12.18)

Here *ijk is defined by:

*ijk =

9::::;

::::<

+1 : ijk even permutation of 123

#1 : ijk odd permutation of 123

0 : ijk some indices equal(12.19)

The inverse of Eq. (12.18) is simply 3:

(i = #1

2*ijk(jk (12.20)

So now we have some understanding for how the average rotation is related to thederivatives of the velocity field.

2Multiplying the right hand side of Eq. (12.18) by a factor (#Det(gij))!12 (see [4] p. 98-99 concern-

ing tensor densities) this is a covariant relation where "ij is minus the dual of "i (see [7] p. 88).3We may guess this and then check it. Alternatively we could multiply Eq. (12.18) by #mij , using

the fact that #mij#ijk = 2$mk

12.3. About deformation 59

12.3 About deformationWe can split the velocity derivative tensor into a symmetric and an antisymmetricpart. We have then to first order in xi:

ui = xj%jui (12.21)

= xj"

1

2(%jui # %iuj) +

1

2(%jui + %iuj)

#(12.22)

= xj ((ij + 'ij) (12.23)

Here we have introduced 'ij as:

'ij =1

2(%jui + %iuj) (12.24)

Now we would like to show that this object is related to deformations (of any shapeconnected to the congruence points). It seems obvious that the most general local mo-tion that is non-deforming corresponds to a rigid rotation (as regards the momentaryvelocities). For this kind of motion we must have (there exists an ! such that):

u = ! " x (12.25)

Obviously in this case the motion due to the average rotation is precisely the motion– and the motion due to the average rotation is uk = (k

jxj . From Eq. (12.23) it thenfollows that we must have xj'ij = 0 for all xj . This can only happen if 'ij = 0. Soit follows that rigid motion implies vanishing 'ij

4. Conversely, given that 'ij = 0, itfollows from Eq. (12.23) that ui = (i

jxi. Since (ij is an antisymmetric tensor it follows

from Eq. (12.14), Eq. (12.16) and Eq. (12.17) that this corresponds to a rigid rotation(with rotation vector given by Eq. (12.16)). So we conclude that the congruence isrigid if and only if 'ij = 0.A simpler argument would be to say that 'i

jxj gives the extra velocity apart fromthat coming from an average (best fit) rotation. Hence the congruence is rigid if andonly if 'ij = 0.

12.4 Back to four-dimensional formalismConnecting the three-dimensional (ij and 'ij to their four-dimensional analogies (µ!

and 'µ! is very simple. In freely falling coordinates locally comoving with the congru-ence, the spatial parts of (µ! and 'µ! precisely equals their three-dimensional analo-gies. Moreover, in these coordinates all time components of (µ! and 'µ! vanishes due

4Alternatively we could just evaluate %iuj + %jui for ui = #ijk"jxk. Here "j is to be treated as a

constant (to the necessary order). Using %ixj = $ij and the antisymmetry of #ijk we readily find that&ij = 0.


to the projection. It follows that the congruence is rigid if and only if 'µ! vanishes,independent of what coordinates we are using. Also in other respects the meaningsare analogous. In particular forming (µ

!!xµ (for some vector !xµ orthogonal to uµ)gives that part of the velocity of the congruence point at the position !xµ (relativeto the inertial system whose origin momentarily comoves with the congruence) thatcan be seen as coming from an average rotation.Concerning the expansion scalar ' it is not hard to realize that this corresponds to

the relative volume increase (of a box of reference points) per unit time. For instance' = 0.1s#1 implies a momentary relative volume increase rate of 10% per second.Notice that we can have ' = 0while for instance having expansion in the x-directionand contraction in the y-direction.Having understood the meaning of 'µ! and ' we understand that the shear tensor

&µ! , as defined by Eq. (12.3), describes that part of the local congruence motion thatis neither due to an (average) rotation nor an (average) expansion.As regards the remaining kinematical invariant aµ, it is simply the acceleration of

the congruence.

12.4.1 The four dimensional analogue to the rotation vectorIn the three-dimensional analysis we saw in Eq. (12.20) how we could relate (i to (ij :

(i = #1

2*ijk(jk (12.26)

We would now like to have a corresponding four-covariant expression. In particularwewould like a tensorial expression for a four vector corresponding to (0, !) in freelyfalling coordinates locally comoving with the congruence. A natural extension of Eq.(12.26) is:

(µ =1

2

1%

gu$*

$µ%#(%# (12.27)

Here g = #Det(g"&) (see [4] p. 98-99 concerning tensor densities). Conversely wehave in analogy with Eq. (12.18):

(µ! =1%

gu$*

$µ!#(# (12.28)

So now we have covariant four-tensor equations relating the rotation tensor to thevector describing the local average rotation.

13Lie transport and Lie-differentiation

To form a covariant derivative of a vector defined along a worldline we need a way oftransporting the vector, according to some law of transport, from one point to anotheralong the worldline so that we can form the difference between the actual vectorand the transported vector (at a single point). This general idea of differentiation isrelevant for papers III-VIII – concerning inertial forces and gyroscope precession.There are a few (standard) means of transporting a vector in general relativity.

There is the ordinary parallel transport where, relative to freely falling coordinates,the components of the vector does not change as we move it along the worldlines.Then there is the Fermi-Walker transport, describing how the spin four-vector istransported along a worldline. Lastly there is the Lie transport. This transport isdefined with respect to a reference congruence, assuming the worldline along whichwe transport the vector is directed along the congruence. In this chapter we give alittle background to the concept of contravariant and covariant Lie differentiation.

13.1 Contravariant Lie differentiationSuppose that we have an arbitrary contravariant vector field uµ. We then choose co-ordinates such that uµ = $µ

0 (assuming the vector field to be reasonably well behavedin the region in question). Note that this procedure has nothing to do with geometryor Lorentz structure. The idea is illustrated in Fig. 13.1.The induced time-slices (although strictly speaking what we are doing need not

have anything to do with time) are uniquely defined by the procedure, given an ar-bitrary initial slice (and labeling of this slice). The labeling of the congruence lines(the streamlines of the vector field) is completely arbitrary. In these coordinates wenow want to transport an arbitrary vector Kµ along a congruence line such that thecoordinate derivative of the vector vanishes, see Fig. 13.2.

61

62 13. Lie transport and Lie-differentiation

Change coordinates

Figure 13.1: Adapting coordinates to a (contravariant) vector field

A

B

Figure 13.2: Transporting a vector in the preferred coordinates

We understand that the transport of this vector is uniquely determined by theprocedure above (how we label the congruence lines does not matter)1. Considernow Kµ to be a field that is transported into itself as outlined above. We have in thecoordinates in question (for the moment we will assume the existence of an affineconnection):

%"uµ = 0 (13.1)("uµ = "µ

"&u& (13.2)Here (µ is the covariant derivative. Also we have in the coordinates in question:

0 = u"%"Kµ (13.3)= u"("Kµ # u""µ

"&K& (13.4)= u"("Kµ # ((&uµ)K& (13.5)

So we have our transport equation in covariant form:u"("Kµ = K"("uµ (13.6)

1Also we may understand that what initial arbitrary time slice we are considering does not mattereither. If we choose a different slicing and draw these slices in the original coordinates as depicted inFig. 13.2, they will be tilted – but the tilt must be the same everywhere along a single congruence line.We may thus understand that the time component of the vector in question will be affected, but thetransport will not be affected.

13.1. Contravariant Lie differentiation 63

Actually we do not need to assume an affine connection. In the particular coordinatesin question we have:

u"%"Kµ = 0

%"uµ = 0

=:::>

:::?+ u"%"Kµ = K"%"uµ (13.7)

The latter equation is through its form such that it holds in all coordinates given thatit holds in one system (do the transformation to another system and we find that theextra, non-tensorial, terms cancel). Assuming then that we have a metrical structureand a covariant derivative, equation Eq. (13.6) follows immediately:Also for a bit of extra intuition, assuming that we have a metrical structure, have

a look at Fig. 13.3.

Kµ

Kµ

uµuµ

Figure 13.3: Transporting a vector Kµ as seen from freely falling coordinates.

It is obvious that the change of Kµ should depend on the change of uµ in thedirection of Kµ. Our first guess from this point of view would likely be:

DKµ

D+= K"("uµ (13.8)

This is in fact exactly Eq. (13.6), so the transport equation is very intuitively reason-able. Also, given an arbitrary vector fieldKµ, we can define the Lie derivative of thisfield as:

LuKµ = u"("Kµ # K"("uµ (13.9)

We can also express this in terms of the kinematical invariants of the congruence. Wehave:

("uµ = (µ" + 'µ

" # aµu" (13.10)In particular, for spatial Kµ, we have then:

LuKµ = u"("Kµ # ((µ

" + 'µ")K" (13.11)


13.2 Covariant Lie differentiationWemay perform a similar analysis considering a covariant vector field Kµ. We anal-ogously demand that the components of this vector should be unaltered as we moveit along the congruence in the preferred coordinates. Analogously to Eq. (13.7) wehave

u"%"Kµ = 0

%"uµ = 0

=:::>

:::?& u"%"Kµ = #K"%µu" (13.12)

We note that we have inserted a minus sign here as compared to Eq. (13.7). Whereasthe equality holds equally well independently of what sign we choose in the pre-ferred coordinates, the sign is chosen such that the equality holds in any coordinatesystem. To prove this we evaluate u!" '

'x!! K !µ+K !

"'

'x!µ u!", using the definitions of howthe covariant and contravariant objects are related to their non-primed versions. Wealso use the trick of differentiating a Kronecker delta in the form:

0 =%

%x!µ

%%x!"

%x$

%x#

%x!"

&

(13.13)

Expanding this derivative and using the resulting expression, we readily find thatu" '

'x! Kµ + K" ''x! uµ is in fact a tensor (although its individual terms are not). Hence

if it vanishes in one system it vanishes in all systems. It follows that a covariant vectorfield Lie-dragged into itself obeys:

u"%"Kµ = #K"%µu" (13.14)

Note that it is not only the sign that differs from the analogous equation Eq. (13.7)of the preceding section. How the summation runs is also different. For this case theform is less intuitive.Lie differentiation of a covariant field is then given by:

LuKµ = u"("Kµ + K"(µu" (13.15)

Note that here we are using covariant differentiation (to make the derivative mani-festly covariant), but the form is such that we could instead use ordinary differentia-tion.Notice in particular that if we have the Lie derivative of a covariant field we do not

get the Lie derivative of the corresponding contravariant field by raising the formerwith the metric:

gµ"LuK" ,= LuKµ (13.16)

In other words Lie-differentiation does not in general commute with the raising andlowering of indices 2.

2If the vector field obeys ("uµ = 0, Lie differentiation does however commute with lowering andraising of indices.

13.3. Comments 65

13.3 Comments

One might consider other transports of the vector in question, not just along the con-gruence, as depicted in Fig. 13.4

A

B

Figure 13.4: Trying a different transport equation with respect to a reference congru-ence.

In a particular set of coordinates we may here demand that the components of thevector should be unaltered as we transport it. For this case it is however obvious thatthe labeling of the congruence lines will affect the transport.To gain intuition concerning the difference between the raised covariant trans-

port and the ordinary contravariant transport, let us study a particular example in2+1-dimensional special relativity. Let Kµ = %µ, where , = x in inertial coordinates(t, x, y). Now consider another coordinate system that is shearing relative to the iner-tial coordinates such that x! = x as depicted in Fig. 13.5.

xµ

x!µ

xx

yy t

t = 0

t = T

Figure 13.5: To the left the shearing coordinates (thin lines) relative to the inertialcoordinates (thick lines) at a time t = T . To the right the shearing coordinates in a2+1 spacetime illustration seen relative to the inertial coordinates.


We have in the inertial coordinates %µ, = (0, 1, 0). Also in the shearing coordinateswe have %!

µ, = (0, 1, 0). Now consider Lie transporting (with respect to the shearingcoordinates) a contravariant vector directed in the x-direction at t = 0 along the con-gruence. At t = T it will be directed in the x!µ direction (see Fig. 13.5). On the otherhand, if we consider a covariant transport of the same vector then in the shearing co-ordinates (0,1,0) will go to (0,1,0) (per definition), but the latter vector – when raisedwith the metric – we know has the meaning of the gradient of , which is directedin the xµ direction. Thus relative to the shearing coordinates the raised covariantlytransported vector will rotate. So here we have an intuitive example illustrating thedifference between the contravariant Lie transport and the raised covariant Lie trans-port.

BIBLIOGRAPHY

[1] Epstein, L. C. (1994). Relativity Visualized, (Insight Press, San Francisco), ch.10,11,12

[2] Marolf, D. (1999), Gen. Rel. Grav. 31, 919

[3] W. Rindler (1994), Am. J. Phys 62, 887-893.

[4] Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of theGeneral Theory of Relativity, (John Wiley & Sons, U.S.A)

[5] Kristiansson, S., Sonego, S., Abramowicz, M.A. (1998). Gen. Rel. Grav. 30, 275

[6] D’Inverno, R. (1998). Introducing Einstein’s Relativity, (Oxford University Press,Oxford), p. 99-101

[7] Misner C.W., Thorne K. S., Wheeler J. A. (1973). Gravitation, (W.H Freeman andCompany, New York), 841

[8] Rindler, W. (2001). Relativity: Special, General and Cosmological, (Oxford Univer-sity Press, Oxford), p. 267-272

[9] Rindler, W. (1977). Essential Relativity: Special, General and Cosmological,(Springer Verlag, New York), 204-207

[10] Hawking, S.W., Ellis, G.F.R. (1973). The large scale structure of space-time, (Cam-bridge University Press, Cambridge), 39

[11] Dray, T. (1989) , Am.J.Phys. 58, 822-825

[12] Bini D, Carini P, Jantzen RT, Proceedings of the Eighth Marcel GrossmannMeeting on General Relativity, Tsvi Piran, Editor, World Scientific, Singapore,A (1998) p. 376-397

[13] Bini D, Carini P, Jantzen RT, Int. Journ. Mod. Phys. D, 6, No 1 (1997) p. 14

[14] Jantzen RT, Carini P, Bini D, Ann. Phys. 215, No 1 (1992) p. 1-50

67

68 BIBLIOGRAPHY

[15] Marek A. Abramowicz, Jean-Pierre Lasota, Class. Quantum Grav., 14 (1997) p.A23-A30

[16] Marek A. Abramowicz, Scientific American (march) 1993

[17] Marek A Abramowicz, Pawel Nurowski and Norbert Wex, Class. QuantumGrav., 12 (1995) p. 1467-1472

[18] Marek A. Abramowicz, Mon. Not. R. astr. Soc., 256 (1992) p. 710-718

[19] Marek A. Abramowicz, Pawel Nurowski and Norbert Wex, Class. QuantumGrav., 10 (1995) p. L183-L186

[20] Perlick V., Class. Quantum Grav., 7 (1990) p. 1319-1331

[21] George E. A. Matsas, Phys. Rev. D., (2003) 68

[22] Rindler W.,Perlick V., (1990). Gen. Rel. Grav. 22 (9), p 1067-1081

[23] Muller R. A., Am. Journ. Phys. (1992) 60 (4), 313

[24] Gravity Probe B Mission web page, http://einstein.stanford.edu/

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